Manchester Memoirs, Vol. liii. (1909), No. \%. 3 



traditional error of Descartes and Newton still survives in 

 manuals of elementary and advanced science, under the 

 name of momentum, in contradiction of Newton's second 

 law which expressly excludes the element of time in the 

 measure of the quantity of motion in a body. 



5. In the preface to his Principia Newton set forth 

 with singular lucidity and ingenuousness, the dependence 

 of the mathematical principles of natural philosophy 

 upon experimental mechanics in their application to the 

 motions of celestial bodies. It will therefore be obvious 

 that the question, whether the quantity of motion in a 

 planetary system is simply as the velocity, or as the 

 square of the velocity is one of fundamental importance. 

 So far as I know, no attempt has yet been made to deal 

 with this problem, and no explication of astronomical 

 science can be considered complete so long as it remains 

 unsolved. 



6. In order to demonstrate that the moving force by 

 which the moon and other celestial bodies are maintained 

 in their orbits is as the square of the velocity, it is 

 postulated as general knowledge in physical astronomy : — 



(rt) That the equatorial circumference of the earth 

 is 24,900 miles. 



{U) That the versed sine of five miles (or more 

 exactly, 4"936 miles) of the earth's circum- 

 ference is 193 inches = 16 feet i inch. 



{c) That a body at rest near the earth's surface falls 

 perpendicularly through the versed sine of 

 4-936 miles of arc of the earth's circum- 

 ference =16 feet I inch during one second 

 of time. 



id) That as versed sines are as the squares of their 

 arcs, and the accelerative force of gravity 

 increases in the same proportion, a body 



